Applied Math Colloquium

Building 2, 190

Speaker: Vladimir Koltchinskii (Georgia Institute of Technology)Title: Estimation of functionals of covariance operators in high-dimensional and infinite-dimensional models.Abstract: In many problems of high-dimensional statistics and machine learning, it is of importance to estimate some low-dimensional features of unknown high-dimensional or infinite-dimensional covariance operators. In particular, the features of interest include various spectral characteristics of the covariance operator such as its eigenvalues, linear forms of its eigenvectors, bilinear forms of its spectral projections, etc. More generally, the features could be represented as locally smooth functionals of the covariance. Naive plug-in estimators based on sample covariance are usually sub-optimal due to their large bias and higher order bias reduction methods are of crucial importance in these problem. We study functional estimation problem in a dimension-free framework with its complexity characterized by so called effective rank of the covariance operator. In this framework, we developed new estimators of a given functional of unknown covariance based on linear aggregation of several plug-in estimators with different sample sizes. We show that these estimators provide higher order bias reduction and achieve the minimax optimal error rates in broad classes of H\"older smooth functionals.